Exercise 2 (5 points)

Part A

A craftsman creates chocolate candies whose shape recalls the profile of the local mountain. The base of such a candy is modeled by the shaded surface defined in an orthonormal coordinate system with unit 1 cm.
This surface is bounded by the x-axis and the graph denoted $\mathscr { C } _ { f }$ of the function $f$ defined on $[ - 1 ; 1 ]$ by: $$f ( x ) = \left( 1 - x ^ { 2 } \right) \mathrm { e } ^ { x } .$$
The objective of this part is to calculate the volume of chocolate needed to manufacture a chocolate candy.

1. a. Justify that for all $x$ belonging to the interval $[ - 1 ; 1 ]$ we have $f ( x ) \geqslant 0$. b. Show using integration by parts that: $$\int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x = 2 \int _ { - 1 } ^ { 1 } x \mathrm { e } ^ { x } \mathrm {~d} x .$$
2. The volume $V$ of chocolate, in $\mathrm { cm } ^ { 3 }$, needed to manufacture a candy is given by: $$V = 3 \times S$$ where $S$ is the area, in $\mathrm { cm } ^ { 2 }$, of the colored surface. Deduce that this volume, rounded to $0.1 \mathrm {~cm} ^ { 3 }$, equals $4.4 \mathrm {~cm} ^ { 3 }$.

Part B

We now consider the profit realized by the craftsman on the sale of these chocolate candies as a function of the weekly sales volume.
This profit can be modeled by the function $B$ defined on the interval $[ 0.01 ; + \infty [$ by: $$B ( q ) = 8 q ^ { 2 } [ 2 - 3 \ln ( q ) ] - 3 .$$
The profit is expressed in tens of euros and the quantity $q$ in hundreds of candies. We admit that the function $B$ is differentiable on $\left[ 0.01 ; + \infty \left[ \right. \right.$. We denote $B ^ { \prime }$ its derivative function.

1. a. Determine $\lim _ { q \rightarrow + \infty } B ( q )$. b. Show that, for all $q \geqslant 0.01 , B ^ { \prime } ( q ) = 8 q ( 1 - 6 \ln ( q ) )$. c. Study the sign of $B ^ { \prime } ( q )$, and deduce the direction of variation of $B$ on $[ 0.01 ; + \infty [$. Draw the complete variation table of function $B$. d. What is the maximum profit, to the nearest euro, that the craftsman can expect?
2. a. Show that the equation $B ( q ) = 10$ has a unique solution $\beta$ on the interval $[1.2; + \infty [$. Give an approximate value of $\beta$ to $10 ^ { - 3 }$ near. b. We admit that the equation $B ( q ) = 10$ has a unique solution $\alpha$ on $[ 0.01 ; 1.2 [$. We are given $\alpha \approx 0.757$. Deduce the minimum and maximum number of chocolate candies to sell to achieve a profit greater than 100 euros.

In the orthonormal coordinate system (O; I, J), we have represented:

• the line with equation $y = x$;
• the line with equation $y = 1$;
• the line with equation $x = 1$;
• the parabola with equation $y = x ^ { 2 }$.

We can thus divide the square OIKJ into three zones.
Part A Prove the results shown in the table below.

ZONE	ZONE 1	ZONE 2	ZONE 3
AREA	$\frac { 1 } { 2 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 6 }$

Part B: a first game
A player throws a dart at the square above. It is admitted that the probability that it lands on a zone is equal to the area of that zone. Thus, the probability that the dart lands on ZONE 3 is equal to $\frac { 1 } { 6 }$.

• If the dart lands on ZONE 3, then the player tosses a fair coin. If the coin lands on HEADS, then the player wins, otherwise he loses.
• If the dart lands on a zone other than ZONE 3, then the player rolls a fair six-sided die. If the die lands on FACE 6, then the player wins, otherwise he loses.

We note the following events: $T$: ``the dart lands on ZONE 3''; $G$: ``the player wins''.

1. Represent the situation with a weighted tree.
2. Prove that the probability of event $G$ is equal to $\frac { 2 } { 9 }$.
3. Given that the player has won, what is the probability that the dart landed on ZONE 3?

Part C: a second game
A player, called player $n^{\mathrm{o}}1$, throws a dart at the previous square. As in Part B, it is admitted that the probability that the dart lands on each of the zones is equal to the area of that zone. The player wins a sum equal, in euros, to the number of the zone. For example, if the dart lands on ZONE 3, the player wins 3 euros. We denote $X _ { 1 }$ the random variable giving the winnings of player $n^{\circ}1$. We denote respectively $E \left( X _ { 1 } \right)$ and $V \left( X _ { 1 } \right)$ the expectation and variance of the random variable $X _ { 1 }$.

1. a. Calculate $E \left( X _ { 1 } \right)$. b. Show that $V \left( X _ { 1 } \right) = \frac { 5 } { 9 }$.
2. A player $n^{\circ}2$ and a player $n^{\circ}3$ play in turn, under the same conditions as player $n^{\circ}1$. It is admitted that the games of these three players are independent of each other. We denote $X _ { 2 }$ and $X _ { 3 }$ the random variables giving the winnings of players $n^{\circ}2$ and $n^{\circ}3$. We denote $Y$ the random variable defined by $Y = X _ { 1 } + X _ { 2 } + X _ { 3 }$. a. Determine the probability that $Y = 9$. b. Calculate $E ( Y )$. c. Justify that $V ( Y ) = \frac { 5 } { 3 }$.

Question 152
A figura mostra um quadrado $ABCD$ com lado de 4 cm. Os pontos $M$ e $N$ são os pontos médios dos lados $AB$ e $CD$, respectivamente.
[Figure]
A área da região sombreada, em cm², é
(A) 4 (B) 6 (C) 8 (D) 10 (E) 12

Question 166
Um triângulo tem lados medindo 7 cm, 24 cm e 25 cm. A área desse triângulo, em cm², é
(A) 42 (B) 84 (C) 87,5 (D) 168 (E) 175

In a right triangle, the hypotenuse measures 10 cm and one of the legs measures 6 cm. What is the area, in square centimeters, of this triangle?
(A) 20
(B) 24
(C) 30
(D) 36
(E) 48

The following is the graph of a continuous function $y = f ( x )$.
When the inverse function $g ( x )$ of function $f ( x )$ exists and is continuous on the interval $[ 0,1 ]$, the limit value $$\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left\{ g \left( \frac { k } { n } \right) - g \left( \frac { k - 1 } { n } \right) \right\} \frac { k } { n }$$ has the same value as which of the following? [4 points]
(1) $\int _ { 0 } ^ { 1 } g ( x ) d x$
(2) $\int _ { 0 } ^ { 1 } x g ( x ) d x$
(3) $\int _ { 0 } ^ { 1 } f ( x ) d x$
(4) $\int _ { 0 } ^ { 1 } x f ( x ) d x$
(5) $\int _ { 0 } ^ { 1 } \{ f ( x ) - g ( x ) \} d x$

Find the area of the region enclosed by the curve $y = 3 \sqrt { x - 9 }$, the tangent line to this curve at the point $( 18,9 )$, and the $x$-axis. [4 points]

(Calculus) Find the length of the curve $y = \frac { 1 } { 3 } \left( x ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } }$ from $x = 0$ to $x = 6$. [4 points]

A quadratic function $f(x)$ with leading coefficient 1 satisfies $f(3) = 0$ and $$\int_{0}^{2013} f(x)\, dx = \int_{3}^{2013} f(x)\, dx$$ If the area enclosed by the curve $y = f(x)$ and the $x$-axis is $S$, find the value of $30S$. [4 points]

For the function $f ( x ) = \frac { 1 } { x }$, what is the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } f \left( 1 + \frac { 2 k } { n } \right) \frac { 2 } { n }$? [3 points]
(1) $\ln 6$
(2) $\ln 5$
(3) $2 \ln 2$
(4) $\ln 3$
(5) $\ln 2$

For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. When $n = 1$, what is the area of the region enclosed by the line segment PQ, the curve $y = f ( x )$, and the $y$-axis? [3 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 19 } { 12 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 7 } { 4 }$
(5) $\frac { 11 } { 6 }$

A quadratic function $f ( x )$ satisfies $f ( 0 ) = 0$ and the following conditions. (가) $\int _ { 0 } ^ { 2 } | f ( x ) | d x = - \int _ { 0 } ^ { 2 } f ( x ) d x = 4$ (나) $\int _ { 2 } ^ { 3 } | f ( x ) | d x = \int _ { 2 } ^ { 3 } f ( x ) d x$ Find the value of $f ( 5 )$. [4 points]

Let region $A$ be enclosed by the curve $y = e ^ { 2 x }$, the $y$-axis, and the line $y = - 2 x + a$, and let region $B$ be enclosed by the curve $y = e ^ { 2 x }$ and the two lines $y = - 2 x + a , x = 1$. When the area of $A$ equals the area of $B$, what is the value of the constant $a$? (Here, $1 < a < e ^ { 2 }$) [3 points]
(1) $\frac { e ^ { 2 } + 1 } { 2 }$
(2) $\frac { 2 e ^ { 2 } + 1 } { 4 }$
(3) $\frac { e ^ { 2 } } { 2 }$
(4) $\frac { 2 e ^ { 2 } - 1 } { 4 }$
(5) $\frac { e ^ { 2 } - 1 } { 2 }$

The area enclosed by the curve $y = - 2 x ^ { 2 } + 3 x$ and the line $y = x$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

What is the area of the region enclosed by the curve $y = e ^ { 2 x }$, the $x$-axis, and the two lines $x = \ln \frac { 1 } { 2 }$ and $x = \ln 2$? [3 points]
(1) $\frac { 5 } { 3 }$
(2) $\frac { 15 } { 8 }$
(3) $\frac { 15 } { 7 }$
(4) $\frac { 5 } { 2 }$
(5) 3

When the line $x = k$ bisects the area enclosed by the curve $y = x ^ { 2 } - 5 x$ and the line $y = x$, what is the value of the constant $k$? [3 points]
(1) 3
(2) $\frac { 13 } { 4 }$
(3) $\frac { 7 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 4

Let $A$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the $y$-axis, and let $B$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the line $x = 2$. When $A = B$, what is the value of the constant $k$? (Here, $4 < k < 5$) [4 points]
(1) $\frac { 25 } { 6 }$
(2) $\frac { 13 } { 3 }$
(3) $\frac { 9 } { 2 }$
(4) $\frac { 14 } { 3 }$
(5) $\frac { 29 } { 6 }$

For the function $f(x) = \frac{1}{9}x(x-6)(x-9)$ and a real number $t$ with $0 < t < 6$, the function $g(x)$ is defined as $$g(x) = \begin{cases} f(x) & (x < t) \\ -(x-t) + f(t) & (x \geq t) \end{cases}$$ Find the maximum area of the region enclosed by the graph of $y = g(x)$ and the $x$-axis. [4 points]
(1) $\frac{125}{4}$
(2) $\frac{127}{4}$
(3) $\frac{129}{4}$
(4) $\frac{131}{4}$
(5) $\frac{133}{4}$

A cubic function $f(x)$ with leading coefficient 1 satisfies $$f(1) = f(2) = 0, \quad f'(0) = -7$$ Let Q be the point where the line segment OP intersects the curve $y = f(x)$ other than P, where O is the origin and $\mathrm{P}(3, f(3))$. Let $A$ be the area enclosed by the curve $y = f(x)$, the $y$-axis, and the line segment OQ, and let $B$ be the area enclosed by the curve $y = f(x)$ and the line segment PQ. What is the value of $B - A$? [4 points]
(1) $\frac{37}{4}$
(2) $\frac{39}{4}$
(3) $\frac{41}{4}$
(4) $\frac{43}{4}$
(5) $\frac{45}{4}$

For the function $f ( x ) = x ^ { 2 } - 4 x - 3$, let $l$ be the tangent line to the curve $y = f ( x )$ at the point $( 1 , - 6 )$, and for the function $g ( x ) = \left( x ^ { 3 } - 2 x \right) f ( x )$, let $m$ be the tangent line to the curve $y = g ( x )$ at the point $( 1,6 )$. What is the area of the figure enclosed by the two lines $l , m$ and the $y$-axis? [4 points]
(1) 21
(2) 28
(3) 35
(4) 42
(5) 49

16. As shown in the figure, a water channel with an isosceles trapezoidal cross-section has its boundary deformed into a parabolic shape due to silt deposition (shown by the dashed line in the figure). The ratio of the original maximum flow to the current maximum flow is $\_\_\_\_$ [Figure]

III. Solution Questions (This section has 6 questions totaling 70 points. Solutions must include explanations, proofs, and calculation steps.)

If the set $\left\{ y \mid y = x + t \left( x ^ { 2 } - x \right) , 0 \leq t \leq 1, 1 \leq x \leq 2 \right\}$ represents a figure where the maximum distance between two points is $d$ and the area is $S$,
A. $d = 3 , S < 1$
B. $d = 3 , S > 1$
C. $d = \sqrt { 10 } , S < 1$
D. $d = \sqrt { 10 } , S > 1$

Draw the missing part of $G _ { h }$ in Figure 2. Sub-task Part A 4b $( 2 \mathrm { marks } )$ Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$.
Sub-task Part A 4c $( 2 \mathrm { marks } )$ State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$.
The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$. [Figure]
(1a) [5 marks] Verify by calculation that $G _ { f }$ is symmetric with respect to the $y$-axis, and investigate the behavior of $f$ for $x \rightarrow + \infty$ using the function term. Determine those $x$-values for which $f ( x ) = 0,96$ holds.
(1b) [4 marks] Investigate by calculation the monotonicity behavior of $G _ { f }$. $\left( \right.$ for verification: $\left. f ^ { \prime } ( x ) = \frac { 4 x } { \left( x ^ { 2 } + 1 \right) ^ { 2 } } \right)$
(1c) [4 marks] Determine by calculation an equation of the tangent $t$ to $G _ { f }$ at the point ( $3 \mid f ( 3 )$ ). Calculate the angle at which $t$ intersects the $x$-axis, and draw $t$ in Figure 1 (Part B).
Now consider the integral function $F : x \mapsto \int _ { 0 } ^ { x } f ( t ) \mathrm { dt }$ defined on $\mathbb { R }$; its graph is denoted by $G _ { F }$. (2a) [5 marks] Justify that $F$ has a zero at $x = 0$, and use the course of $G _ { f }$ to make plausible that there is another zero of $F$ in the interval [ $1 ; 3$ ]. State what special property $G _ { F }$ has at the point $( - 1 \mid F ( - 1 ) )$, and justify your statement.
(2b) [2 marks] The line with equation $y = x - 1$ bounds a triangle together with the coordinate axes. State the area of this triangle and the resulting approximate value for $F ( 1 )$.
(2c) [5 marks] Figure 2 (Part B) shows the graph $G _ { f }$ and the graph $G _ { g }$ of the function $g : x \mapsto - \cos \left( \frac { \pi } { 2 } x \right)$ defined on $\mathbb { R }$. Describe how $G _ { g }$ is obtained from the graph of the function $x \mapsto \cos x$ defined on $\mathbb { R }$, and calculate another approximate value for $F ( 1 )$ by integrating $g$.
[Figure]
Fig. 2 (Part B)
(for verification: $F ( 1 ) \approx - \frac { 2 } { \pi }$ )
(2d) [4 marks] Calculate the arithmetic mean of the two approximate values computed in tasks 2b and 2c. Sketch the graph of $F$ for $0 \leq x \leq 3$ taking into account the previous results in Figure 1 (Part B).
For each value $k > 0$, the points $P _ { k } ( - k \mid f ( - k ) )$ and $Q _ { k } ( k \mid f ( k ) )$ lying on $G _ { f }$ together with the point $R ( 0 \mid 1 )$ determine an isosceles triangle $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$.
(3a) [5 marks] Calculate the area of the corresponding triangle $\mathrm { P } _ { 2 } \mathrm { Q } _ { 2 } \mathrm { R }$ for $k = 2$ (see Figure 3). Then show that the area of the triangle $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$ in general can be described by the term $A ( k ) = \frac { 2 k } { k ^ { 2 } + 1 }$. [Figure]
(3b) [6 marks] Show that there is a value of $k > 0$ for which $A ( k )$ is maximal. Calculate this value of $k$ and the area of the corresponding triangle $\mathrm { P } _ { k } \mathrm { Q } _ { k } \mathrm { R }$.

A function $f$ is given by the function equation $f ( x ) = x ^ { 3 } - 3 x , x \in \mathbb { R }$.
The graph of $f$ is shown in the figure.
(1) Justify that the graph of $f$ is point-symmetric about the origin.
(2) The graph of $f$ encloses an area with the $x$-axis in the second quadrant. Determine the area $A$ of this region by calculation.
[For verification: $A = 2.25 \mathrm { FE }$.]
(3) Given is the line $g : y = - 2 x , x \in \mathbb { R }$. Determine the ratio in which the line $g$ divides the area from (2).

(1) Sketch the graph of $f _ { 0 }$ in the interval [-2;2] in the figure. (2) The graph of $f _ { 0 }$ encloses an area with the $x$-axis in the second quadrant.
Determine the area $A _ { 0 }$ of this region by calculation. [For verification: $A _ { 0 } = 2,25 \mathrm { FE }$.] (3) Given is the line $g : y = - 2 x , x \in \mathbb { R }$.
Determine the ratio in which the line $g$ divides the area from (2).